Editing Dirichlet distribution (section)

===Moments===
Let <math>X = (X_1, \ldots, X_K)\sim\operatorname{Dir}(\boldsymbol\alpha)</math>.

Let

<math display=block>\alpha_0 = \sum_{i=1}^K \alpha_i.</math>

Then<ref>Eq. (49.9) on page 488 of [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471183873.html Kotz, Balakrishnan & Johnson (2000). Continuous Multivariate Distributions. Volume 1: Models and Applications. New York: Wiley.]</ref><ref>{{cite book|author=BalakrishV. B.|year=2005|title=A Primer on Statistical Distributions|publisher=John Wiley & Sons, Inc.|location=Hoboken, NJ|isbn=978-0-471-42798-8|chapter="Chapter 27. Dirichlet Distribution"|page=[https://archive.org/details/primeronstatisti0000bala/page/274 274]|chapter-url=https://archive.org/details/primeronstatisti0000bala/page/274}}</ref>

<math display=block>\operatorname{E}[X_i] = \frac{\alpha_i}{\alpha_0},</math>
<math display=block>\operatorname{Var}[X_i] = \frac{\alpha_i (\alpha_0-\alpha_i)}{\alpha_0^2 (\alpha_0+1)}.</math>

Furthermore, if <math> i\neq j</math>

<math display=block>\operatorname{Cov}[X_i,X_j] = \frac{- \alpha_i \alpha_j}{\alpha_0^2 (\alpha_0+1)}.</math>

The covariance matrix is [[invertible matrix|singular]].

More generally, moments of Dirichlet-distributed random variables can be expressed in the following way. For <math> \boldsymbol{t}=(t_1,\dotsc,t_K) \in \mathbb{R}^K</math>, denote by <math>\boldsymbol{t}^{\circ i} = (t_1^i,\dotsc,t_K^i)</math> its {{mvar|i}}-th [[Hadamard product (matrices)#Analogous operations|Hadamard power]]. Then,<ref>{{Cite journal |last=Dello Schiavo |first=Lorenzo |date=2019 |title=Characteristic functionals of Dirichlet measures |journal=Electron. J. Probab. |volume=24 |pages=1–38 |doi=10.1214/19-EJP371 |doi-access=free|arxiv=1810.09790 }}</ref>

<math>\operatorname{E}\left[ (\boldsymbol{t} \cdot \boldsymbol{X})^n \right] = \frac{n! \, \Gamma ( \alpha_0 )}{\Gamma (\alpha_0+n)} \sum \frac{{t_1}^{k_1} \cdots {t_K}^{k_K}}{k_1! \cdots k_K!} \prod_{i=1}^K \frac{\Gamma(\alpha_i + k_i)}{\Gamma(\alpha_i)} = \frac{n! \, \Gamma ( \alpha_0 )}{\Gamma (\alpha_0+n)} Z_n(\boldsymbol{t}^{\circ 1} \cdot \boldsymbol{\alpha}, \cdots, \boldsymbol{t}^{\circ n} \cdot \boldsymbol{\alpha}),</math>

where the sum is over non-negative integers <math>k_1,\ldots,k_K</math> with <math>n=k_1+\cdots+k_K</math>, and <math>Z_n</math> is the [[Cycle index#Symmetric group Sn|cycle index polynomial]] of the [[Symmetric group]] of degree {{mvar|n}}.

We have the special case <math>\operatorname{E}\left[ \boldsymbol{t} \cdot \boldsymbol{X} \right] = \frac{\boldsymbol{t} \cdot \boldsymbol{\alpha}}{\alpha_0}.  </math>

The multivariate analogue <math display="inline">\operatorname{E}\left[ (\boldsymbol{t}_1 \cdot \boldsymbol{X})^{n_1} \cdots (\boldsymbol{t}_q \cdot \boldsymbol{X})^{n_q} \right]</math> for vectors <math>\boldsymbol{t}_1, \dotsc, \boldsymbol{t}_q \in \mathbb{R}^K</math> can be expressed<ref>{{ cite arXiv | last1=Dello Schiavo | first1=Lorenzo | last2=Quattrocchi | first2=Filippo | date=2023 | title=Multivariate Dirichlet Moments and a Polychromatic Ewens Sampling Formula | eprint=2309.11292 | class=math.PR }}</ref> in terms of a color pattern of the exponents <math>n_1, \dotsc, n_q</math> in the sense of [[Pólya enumeration theorem]].

Particular cases include the simple computation<ref>{{cite web|last1=Hoffmann|first1=Till|title=Moments of the Dirichlet distribution|url=https://tillahoffmann.github.io/Moments-of-the-Dirichlet-distribution/|archive-url=https://web.archive.org/web/20160214015422/https://tillahoffmann.github.io/Moments-of-the-Dirichlet-distribution/ |access-date=14 February 2016|archive-date=2016-02-14 }}</ref>

<math display=block>\operatorname{E}\left[\prod_{i=1}^K X_i^{\beta_i}\right] = \frac{B\left(\boldsymbol{\alpha} + \boldsymbol{\beta}\right)}{B\left(\boldsymbol{\alpha}\right)} = \frac{\Gamma\left(\sum\limits_{i=1}^K \alpha_{i}\right)}{\Gamma\left[\sum\limits_{i=1}^K (\alpha_i+\beta_i)\right]}\times\prod_{i=1}^K \frac{\Gamma(\alpha_i+\beta_i)}{\Gamma(\alpha_i)}.</math>